Smoothstep

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A plot of the smoothstep(x) and smootherstep(x) functions.

Smoothstep is a scalar interpolation function commonly used in computer graphics[1][2] and video game engines.[3] The function interpolates smoothly between two input values based on a third one that should be between the first two. The returned value is clamped between 0 and 1.

The slope of the smoothstep function tends toward zero at both edges. This makes it easy to create a sequence of transitions using smoothstep to interpolate each segment rather than using a more sophisticated or expensive interpolation technique.

As pointed out in MSDN and OpenGL documentation, smoothstep implements cubic Hermite interpolation after doing a clamp:

 \operatorname{smoothstep}(t) = 3t^2 - 2t^3

An example implementation provided by AMD[4] follows.

float smoothstep(float edge0, float edge1, float x)
{
    // Scale, bias and saturate x to 0..1 range
    x = clamp((x - edge0)/(edge1 - edge0), 0.0, 1.0); 
    // Evaluate polynomial
    return x*x*(3 - 2*x);
}

Variations[edit]

Ken Perlin suggests[5] an improved version of the smoothstep function which has zero 1st and 2nd order derivatives at t=0 and t=1:

 \operatorname{smootherstep}(t) = 6t^5 - 15t^4 + 10t^3

Reference implementation:

float smootherstep(float edge0, float edge1, float x)
{
    // Scale, and clamp x to 0..1 range
    x = clamp((x - edge0)/(edge1 - edge0), 0.0, 1.0);
    // Evaluate polynomial
    return x*x*x*(x*(x*6 - 15) + 10);
}

Origin[edit]

3rd order equation[edit]

We start with a generic third order polynomial function and its first derivative:

\begin{alignat}{9}
f(t) &&\; = \;&&   a_3 t^3 &&\; + \;&& a_2 t^2 &&\; + \;&& a_1 t &&\; + \;&& a_0 & \\
f'(t) &&\; = \;&& 3 a_3 t^2 &&\; + \;&& 2 a_2 t &&\; + \;&& a_1 &
\end{alignat}

Applying the desired values for the function at both endpoints we get:

\begin{alignat}{13}
f(0) &&\; = \;&& 0 \;\;\;\;\;&& \Rightarrow &&\;\;\;\;\;   0 \;&& + &&\;   0 \;&& + &&\;   0 \;&& + &&\; a_0 &&\; = \;&& 0 & \\
f(1) &&\; = \;&& 1 \;\;\;\;\;&& \Rightarrow &&\;\;\;\;\; a_3 \;&& + &&\; a_2 \;&& + &&\; a_1 \;&& + &&\; a_0 &&\; = \;&& 1 &
\end{alignat}

Applying the desired values for the first derivative of the function at both endpoints we get:

\begin{alignat}{11}
f'(0) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\;     0 \;&& + &&\;     0 \;&& + &&\; a_1 \;&& = \;&& 0 & \\
f'(1) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\; 3 a_3 \;&& + &&\; 2 a_2 \;&& + &&\; a_1 \;&& = \;&& 0 &
\end{alignat}

Solving the system of 4 unknowns formed by the last 4 equations we obtain the values of the polynomial coefficients:

a_0 = 0 , \;\;\;\;\;\; a_1 = 0 , \;\;\;\;\;\; a_2 = 3 , \;\;\;\;\;\; a_3 = -2

Introducing these coefficients back into the first equation gives the third order smoothstep function:

f(t) = -2t^3 + 3t^2

5th order equation[edit]

We start with a generic fifth order polynomial function, its first derivative and its second derivative:

\begin{alignat}{13}
f(t)   &&\; = \;&&    a_5 t^5 &&\; + \;&&    a_4 t^4 &&\; + \;&&   a_3 t^3 &&\; + \;&&   a_2 t^2 &&\; + \;&& a_1 t &&\; + \;&& a_0 & \\
f'(t)  &&\; = \;&&  5 a_5 t^4 &&\; + \;&&  4 a_4 t^3 &&\; + \;&& 3 a_3 t^2 &&\; + \;&& 2 a_2 t   &&\; + \;&& a_1   & \\
f''(t) &&\; = \;&& 20 a_5 t^3 &&\; + \;&& 12 a_4 t^2 &&\; + \;&& 6 a_3 t   &&\; + \;&& 2 a_2     &
\end{alignat}

Applying the desired values for the function at both endpoints we get:

\begin{alignat}{17}
f(0) &&\; = \;&& 0 \;\;\;\;\;&& \Rightarrow &&\;\;\;\;\;   0 \;&& + &&\;   0 \;&& + &&\;   0 \;&& + &&\;   0 \;&& + &&\;   0 \;&& + &&\; a_0 &&\; = \;&& 0 & \\
f(1) &&\; = \;&& 1 \;\;\;\;\;&& \Rightarrow &&\;\;\;\;\; a_5 \;&& + &&\; a_4 \;&& + &&\; a_3 \;&& + &&\; a_2 \;&& + &&\; a_1 \;&& + &&\; a_0 &&\; = \;&& 1 &
\end{alignat}

Applying the desired values for the first derivative of the function at both endpoints we get:

\begin{alignat}{15}
f'(0) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\;     0 \;&& + &&\;     0 \;&& + &&\;     0 \;&& + &&\;     0 \;&& + &&\; a_1 \;&& = \;&& 0 & \\
f'(1) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\; 5 a_5 \;&& + &&\; 4 a_4 \;&& + &&\; 3 a_3 \;&& + &&\; 2 a_2 \;&& + &&\; a_1 \;&& = \;&& 0 &
\end{alignat}

Applying the desired values for the second derivative of the function at both endpoints we get:

\begin{alignat}{15}
f''(0) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\;      0 \;&& + &&\;      0 \;&& + &&\;     0 \;&& + &&\; 2 a_2 \;&& = \;&& 0 & \\
f''(1) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\; 20 a_5 \;&& + &&\; 12 a_4 \;&& + &&\; 6 a_3 \;&& + &&\; 2 a_2 \;&& = \;&& 0 &
\end{alignat}

Solving the system of 6 unknowns formed by the last 6 equations we obtain the values of the polynomial coefficients:

a_0 = 0 , \;\;\;\;\;\; a_1 = 0 , \;\;\;\;\;\; a_2 = 0 , \;\;\;\;\;\; a_3 = 10 , \;\;\;\;\;\; a_4 = -15 , \;\;\;\;\;\; a_5 = 6

Introducing these coefficients back into the first equation gives the fifth order smootherstep function:

f(t) = 6t^5 - 15t^4 + 10t^3

References[edit]

External links[edit]